PSI - Issue 46

T.L. Castro et al. / Procedia Structural Integrity 46 (2023) 105–111

110

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TL Castro et al. / Structural Integrity Procedia 00 (2019) 000–000

�� � �� � �� �� ���� � � �� � � ��� �� ,

(19)

where

��� ��� �� � �� ��√3 .

(20) An important advantage of the Papadopoulos model refers to the fact that it is independent of critical plane, but it is also to be remembered that its validity is restricted to materials for which range 1⁄√3��� �� �� ⁄ ���� (Papadopoulos et al., 1997). 3.1. Discussing the error index Examining the histograms shown in Fig. 4, one can conclude that the predictions made by the criteria in question exhibit different degrees of dispersion. As listed in Table 3, the error index varies from -38% to 10% for the L&M criterion, indicating a dispersion range of 48%. In a decreasing order, this is followed by a dispersion of 47%, 45%, 41%, 38% e 36% for the S&L, M, Modified C&S, F and C&S criteria, respectively. As to the Papadopoulos criterion, an error index dispersion of 32% is noticeable.

Table 3. Frequency histogram deviations. Average

Std deviation

Min

Median

Max 24.02 22.41 8.78 5.90 10.29 16.68 35.15

Amplitude

2.45 5.30 -2.84 -5.66 -4.36 1.80 3.64

7.99 7.49 7.40 8.26 9.30 5.27 7.53

-21.65 -16.13 -26.81 -35.96 -38.55 -15.34 -12.51

2.85 4.40 -1.56 -3.58 -2.66 2.50 2.90

45.67 38.54 35.59 41.86 48.84 32.02 47.66

Matake, M Findley, F

Carpinteri & Spagnoli, C&S

Modified C&S

Liu & Mahadevan, L&M

Papadopoulos, P

Susmel & Lazzarin, S&L

Table 3 also indicates that, among all of the criteria in question, the Papadopoulos criterion displays the lowest error index average value (situated at 1.8%), with lowest standard deviation (5.2%). However, a low average value of is not a definite indication of good predictive capability as positive values cancel out with negative ones. An effective method of evaluating the efficacy of a given criterion is to determine the number of tests (loading conditions) which belong to a selected interval around zero, for example from -10% to 10%. The closer this number to the total number of tests, the higher will be the predictive capability of the criterion. The percentage of tests belonging to that interval (±10%), as related to the total number of tests, vary between 72% for the Findley criterion to 86% for the C&S’. On the other hand, this percentage amounts to 95% for the Papadopoulos criterion. Percentages obtained for all the criteria are presented in Fig. 5, together with those calculated for an range between -5% and 5%. Again, there is more tests within this range for the Papadopoulos criterion in comparison with all the others.

Fig. 5. Percentage of the tests belonging to central I range around 0%.